alignment-free, self-calibrating elbow angles measurement ......kinematics from imu data, is that...

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This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/JBHI.2016.2639537, IEEE Journal ofBiomedical and Health Informatics

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Alignment-Free, Self-Calibrating Elbow AnglesMeasurement using Inertial Sensors

Philipp Muller1, Marc-Andre Begin2, Thomas Schauer1 and Thomas Seel1

Abstract—Due to their relative ease of handling and low cost,inertial measurement unit (IMU)-based joint angle measurementsare used for a widespread range of applications. These includesports performance, gait analysis and rehabilitation (e.g. Parkin-son’s disease monitoring or post-stroke assessment). However,a major downside of current algorithms, recomposing humankinematics from IMU data, is that they require calibrationmotions and/or the careful alignment of the IMUs with respectto the body segments. In this article, we propose a new method,which is alignment-free and self-calibrating using arbitrarymovements of the user and an initial zero reference arm pose.The proposed method utilizes real-time optimization to identifythe two dominant axes of rotation of the elbow joint. Theperformance of the algorithm was assessed in an optical motioncapture laboratory. The estimated IMU-based angles of a humansubject were compared to the ones from a marker-based opticaltracking system. The self-calibration converged in under 9.5 son average and the RMS errors with respect to the opticalreference system were 2.7◦ for the flexion/extension and 3.8◦ forthe pronation/supination angle. Our method can be particularlyuseful in the field of rehabilitation, where precise manual sensor-to-segment alignment as well as precise, predefined calibrationmovements are impractical.

I. INTRODUCTION

LONGTIME considered as the gold standard in the fieldof human motion measurements for their high degree

of accuracy, optical tracking systems have proven to be poorcandidates in such areas as the remote follow-up on trainingschemes for patient rehabilitation due to their low mobility,complex setup and high cost. For such applications where itis impractical for human subjects to travel to specialized lab-oratories equipped with optical measurement systems, inertialmeasurement units (IMUs) are preferred over exoskeletons andgoniometers because the latter tend to restrict natural motionof the patient and the cost of the former remains low incomparison [1], [2]. In general, an IMU captures the informa-tion provided by a multi-axis accelerometer and a multi-axisgyroscope in order to infer the position and orientation of thesensor by sensor fusion [3], [4]. To compensate for drifts inthe orientation heading inherent to IMUs, some commerciallyavailable systems incorporate a magnetometer into the design,in which case the orientation of the IMU can be known withrespect to a world reference frame.

From the absolute orientation of two IMUs placed on twodifferent body segments forming a joint (e.g. the elbow joint

1Control Systems Group (Fachgebiet Regelungssysteme), Technische Uni-versitat Berlin, 10623 Berlin, Germany{Mueller/Schauer/Seel}@control.TU-Berlin.de

2Universite de Sherbrooke, 2500, Boul. de l’Univ., Sherbrooke, [email protected]

Pronation/SupinationAngle

Flexion/ExtensionAngle

x

yz

Pronation/Supination Axis

Flexion/ExtensionAxis

xyz

Fig. 1: Experimental Setup: Two wireless IMUs are attached to the upper- andthe forearm. Optical markers are placed on the arm and shoulder as part ofthe optical reference system. The IMU coordinate frames are shown as wellas the approximate location of the joint axes.

Frame A

IMU 1Frame A

ωAIMU1

aAIMU1mA

IMU1

RAW

World Frame

Frame B

IMU 2Frame B

ωBIMU2

aBIMU2mB

IMU2

RBW

aA

bB

Fig. 2: A schematic of the elbow joint model. Two segments are connectedby two revolute joints, allowing two degrees of freedom. The rotation axisa is fixed in the frame of IMU 1 whereas the rotation axis b is fixed inthe frame of IMU 2. Both IMUs are arbitrary attached to the segments. Bymeasuring angular velocity, acceleration and magnetic field in the local frame,the absolute orientations RA

W ans RBW of each IMU can be estimated.

depicted in Fig. 1), the angles of this joint can be inferredby decomposing the relative orientation between both IMUsalong the rotation axes of the joint [5]. This process requires,however, the joint rotation axes to be known in each of theIMUs reference frames. As of now, this has been done usingeither careful alignments [6]–[8] and identification of relevantbody landmarks recommended by the International Societyof Biomechanics [5] or by using a set of predefined andprecisely carried out calibration motions [9]–[12]. Automatic



2

orientation calibration is already existent in the field of offlinebody motion capturing. Here, joint angles as well as IMUalignment could be successfully estimated even without the useof magnetometers [13], [14]. However, due to the complexityof the optimization problems, an online measurement was notyet possible.

The purpose of this paper is to propose a new algorithm thatallows the online measurement of the flexion/extension as wellas the pronation/supination angle of the elbow using IMUs.The key merits of the proposition are that neither alignment ofthe sensors nor any specific calibration motions are necessary.This algorithm uses kinematic constraints of the elbow toestimate the dominant axes of rotation of the joint expressedin each IMU reference frame, similarly to what has been donepreviously for the knee joint [15].

A preliminary version of this work has been reported in[16] where a first version of the algorithm was evaluated ona mechanical device. The scope of this article is to give a de-tailed description of the method (including the full derivation,details on the implementation and discussion of the tuning ofparameters), extend the method and most importantly, evaluateit on the human subject.

II. METHOD

A. Design of the Algorithm

The human elbow can be modeled as a two-dimensionaljoint connecting two segments where the two dimensionsare represented by the flexion/extension (FE) and the prona-tion/supination (PS) axes [5]. Two dimensions are sufficientsince the carrying angle is not a degree of freedom [17].

The addition theorem for angular velocities implies, thatthe relative angular velocity of two objects can be decomposedinto a sum of angular velocities around different rotation axes.Applied to a two-dimensional joint, this means that the relativeangular velocity of the joint is a linear combination of its tworotation axes.

Given two IMUs where IMU1 measures in frame A and isattached to segment 1 and IMU2 measures in frame B and isattached to segment 2 (see Fig. 2), then the relative angularvelocity of the elbow in the frame of IMU1 is

ωAr,k = −ωA

IMU1,k + RBA,kωBIMU2,k, (1)

where ωAIMU1,k and ωB

IMU2,k are the angular velocities mea-sured by IMU1 and IMU2, respectively. RBA,k is the rotationmatrix that rotates from frame B into frame A, which can beobtained from the two estimated absolute IMU orientations. kis the index of the time-discrete measurements.

For a joint with two rotational degrees of freedom aroundthe unit axis aA in the frame of IMU1 and bB in the frameof IMU2, the relative angular velocity can also be expressedas

ωAr,k = αkaA + βkRBA,kbB + ek, (2)

where αk and βk are scalars and ek expresses the error in thecase that aA, bB , αk or βk are not known perfectly.

Given that aA and bB are known but do not necessarilyrepresent the true rotation axes, a combination of αk and

βk can be found that minimizes the magnitude of ek. WhenaA 6= ±RBA,kbB the problem is linear and convex1. Hence,the unique optimal solution can be computed with the Moore-Penrose pseudoinverse[

αkβk

]=(MTk Mk

)−1MTkω

Ar,k, (3)

where Mk =[aA RBA,kbB

].

(3) can be rearranged to

αk =(aA)T − (aA)TRBA,kbB(bB)T (RBA,k)T

1− (aA)TRBA,kbB(bB)T (RBA,k)T aAωA

r,k (4)

and

βk =(RBA,kbB)T − (aA)TRBA,kbB(aA)T

1− (aA)TRBA,kbB(bB)T (RBA,k)T aAωA

r,k. (5)

ωAr,k and RBA,k can be determined by using the mea-

surements. Hence, by inserting (4) and (5) in (2), ek canbe expressed using only two unknowns aA and bB . Themagnitude of the error ek is proportional to the magnitudeof ωA

r,k. In order to obtain a term that expresses the errorof the estimated axes and is independent of the magnitude ofωA

r,k, a normalized error en,k is defined as

en,k =

αkaA + βkRBA,kbB −ωA

r,k

‖ωAr,k‖2

if ‖ωAr,k‖2 > 0

0 if ‖ωAr,k‖2 = 0

. (6)

where ‖ · ‖2 is the Euclidean norm.Using the normalized error, a quadratic cost function can

be defined for N measurement samples

J(aA,bB) =1

N

N∑k=1

eTk ek(ωA

r,k)TωA

r,k, ∀ ‖ωA

r,k‖2 > 0. (7)

The two unknown axes aA and bB are unit axes perdefinition and can therefore be reduced to four dimensionsby expressing them in spherical coordinates

aA =

sin θa cos ρasin θa sin ρacos θa

, bB =

sin θb cos ρbsin θb sin ρbcos θb

. (8)

The unknown axes estimate can then be expressed by

φ =[θa ρa θb ρb

]T. (9)

This means, that the nonlinear cost function J has fourunknowns. In order to find a local minimum of this function,it is beneficial to calculate its gradient. The gradient of J withrespect to φ can be expressed as

∂J(aA,bB)∂φ

=2

N

N∑k=1

eTk∂∂φek

(ωAr,k)

TωAr,k. (10)

1aA = ±RBA,kbB would mean that both rotation axes lie on top of

each other (i.e. the joint is one dimensional), in this case the Moore-Penrosepseudoinverse supplies one of the infinite set of optimal solutions



3

By inserting (2) and rearranging we get

∂J(aA,bB)∂φ

= − 2

N

N∑k=1

(∂αk∂φ

eTk aA + αkeTk∂aA

∂φ

+∂βk∂φ

eTk RBA,kbB + βkeTk RBA,k∂bB

∂φ

)1

(ωAr,k)

TωAr,k. (11)

Due to the definition of αk and βk in (3), the error ek standsperpendicular to the plane of aA and RBA,kbB for all k.2 Hence,eTk aA = 0 and eTk RBA,kbB = 0, which simplifies (11) to

∂J(aA,bB)∂φ

= − 2

N

N∑k=1

αkeTk∂aA∂φ + βkeTk RBA,k ∂bB

∂φ

(ωAr,k)

TωAr,k

. (12)

The partial derivatives of the axes aA and bB can be expressedas

∂aA

∂θa=

cos θa cos ρacos θa sin ρa− sin θa

, ∂aA

∂ρa=

− sin θa sin ρasin θa cos ρa

0

, (13)

∂bB

∂θb=

cos θb cos ρbcos θb sin ρb− sin θb

, ∂bB

∂ρb=

− sin θb sin ρbsin θb cos ρb

0

(14)

and∂aA

∂θb=∂aA

∂ρb=∂bB

∂θa=∂bB

∂ρa= 0. (15)

A common way to find a local minimum of a nonlinearcost function is the gradient descent algorithm. Meaning thatan initial guess of φ (given enough iteration steps and a smallenough step size) will lead to a locally optimal estimate of thejoint axes aA and bB .

The joint angles can then be derived from the estimatedaxes and the relative orientation of the two IMUs. The twoestimated rotation axes are possibly nonorthogonal, whichimpedes use of a conventional Euler decomposition. This canbe the case when the solution did not yet converge or when themechanical joint is nonorthogonal. In [18] the Euler decom-position method was generalized for arbitrary nonorthogonalrotation axes. Using RBA,k, aA, bB and aA×bB

‖aA×bB‖2 as a floatingaxis, the relative joint angles can be calculated.

B. Implementation of the Algorithm

For the sake of real-time capability, a window of measure-ments is chosen such that the cost function is calculated forthe last M measurements. With each new measurement, thewindow is shifted by one sample and a step along the newgradient is taken. The rotation matrices RAW and RBW areobtained by using a real-time orientation estimation algorithm.

At a defined point in time, a ”zero pose” (e.g. straight arm)has to be performed; thus, defining the obtained joint anglesto be zero at this arm position. The relative IMU orientationRBA,0 of this pose is stored and with each update of theestimated axes, the current angle offset is calculated using the

2αkaA and βkRBA,kbB define a plane and the shortest vector connecting

ωAr,k and the plane has to be normal to the plane.

ωAr,k

RBA,k aAk

bBk

RBA,0

Flexion/Extension

Parameters:Stepsize StartStepsizeJLP Thresholdα Thresholdβ Threshold

GradientDescentAlgorithm

EulerDecompo-sition

EulerDecompo-sition

Pronation/Supination

Fig. 3: The schematic shows the discrete implementation of the algorithmas well as the generalized Euler decomposition. By estimating the jointangles of the “zero pose”, absolute joint angles can be derived by subtractingthose angles from the current estimate of the relative joint angles. The axisestimation algorithm can be adjusted by the five depicted parameters.

generalized Euler decomposition. The resulting offset is thensubtracted from the calculated angles (see Fig. 3).

To reduce the convergence time as much as possible, arelatively big step size should be chosen. This however canlead to a very unsteady axes estimation due to skin motionartifacts. Skin motion artifacts are movements caused by theelasticity of the soft tissue in between the rigid bone and thesensor. To average out these effects, either a much smallerstep size has to be chosen or the window size has to bebigger than the period of those effects. A large window sizeincreases the computational burden and thus compromises thereal-time capability of the algorithm. To avoid the step sizetradeoff, a two-stage solution is proposed. In the first stage,a bigger step size is chosen. As soon as the cost functionreaches a certain threshold, indicating a good estimate, thestep size is switched to a smaller value. To increase therobustness of the switchover, the cost function J is lowpass-filtered using a discrete Butterworth filter, yielding JLP. Thetwo-stage solution assures a fast convergence without makingthe estimated axes vary over time due to distortions from theskin motion artifacts.

Due to the singularity of the spherical coordinate systemat θ = i(180◦), i ∈ Z, the gradients ∂aAk

∂ρaand ∂bB

k

∂ρb,

respectively, become zero. This renders the gradient (12) tosolely point in the θ direction, which can slow down theconvergence significantly. To avoid this problem, frame A andB, respectively are rotated 90◦ around the x axis as soon as θaor θb is closer than 45◦ to singularity. Naturally, after takingthe gradient step, the estimated axes are rotated back to theirinitial frame.

Given that the rotation axes are estimated correctly, whenno rotation occurs around one of the axes, naturally thegradient for this axis is zero. However, there is always somemeasurement noise and other imperfections present. In orderto avoid an axis from drifting when no motion is applied, athreshold for every axis was introduced. α and β express themagnitudes of the angular velocities around the axes a and b.Thus, when α or β are smaller than the defined threshold, thegradient of the respective axis is set to zero.

C. Experimental Setup

An optical marker-based system (Vicon Motion SystemsLtd. UK) was used as reference and two IMUs for the



4

x

yz

x

yz

Fig. 4: The second, more arbitrary positioning of the IMUs. The orientation aswell as the as the position on the segment was changed to verify the generalityof the proposed method.

measurement. The used IMUs were Xsens MTw wireless units(Xsens Technologies B.V., Netherlands).

As depicted in Fig. 1, the first IMU was placed on theupper arm and the second on the wrist. The first IMU waspositioned as close as possible toward the elbow joint to avoidadditionally measuring internal or external shoulder rotation.Internal or external shoulder rotation would introduce a thirddegree of freedom, rendering a good estimation of a two-dimensional joint impossible.

Eight markers were placed on the subject to capture the FEand PS angle using marker positions. Following the ISB rec-ommendations of joint coordinate systems [5], the followingbony landmarks where used: the Art. Acromioclavulare (AC),Trigonum Scapulae (TS), Angulus Inferior (AI), AngulusAcromialis (AA), Lateral Epicondyle (EL), Medial Epicondyle(EM), Radial Styloid (RS), and the Ulnar Styloid (US). Theselandmarks allow the measurement of the shoulder, upper andlower arm coordinate system.

For the determination of the FE angle, the position of theGlenohumeral rotation center (GH) is estimated [5]. A way toestimate its position is by fitting a sphere to the trajectory ofthe elbow joint. To consider the movements and rotations ofthe shoulder, the trajectory of a marker on the shoulder wassubtracted from the trajectory of the elbow. Subsequently thisrelative elbow position was rotated with the inverse rotation ofthe shoulder at each time. The sphere was fitted to the obtained“statical” elbow trajectory. The estimated center of this spherewas transformed back to the absolute coordinates to obtain theGH estimate.

Using the estimated GH and the measured marker trajecto-ries, the coordinate systems of the upper and lower arm can befound and thus the relative rotation matrix can be calculated.Equivalent to the IMU angle calculation, the joint angles wherethen calculated using the generalized Euler decomposition[18].

The evaluation was done with one healthy male subject(age 25 years). The trials have been approved by the ethicscommittee of the Berlin Chamber of Physicians (ArztekammerBerlin, Eth-25/15). The procedure of the experiment was thefollowing. During each trial, the arm was first held approxi-mately still at a defined position for ten seconds. The ”zeropose” was set in the middle of this still period and thealgorithm was started at the end of it. The subject was askedto do a defined repeating motion. The motion consisted of

TABLE I: The set of parameters that was used for all presented trials includingthe initial axes estimate. In the beginning, the start step size is used. As soonas the filtered cost function hits the error threshold, the smaller step size isused. Steps are only taken for axis aA if α is above the minimum and for bB

if β is above the minimum. Angular velocities below the minimum thresholdare ignored.

Parameter Value

Moving Window Size 100

Step Size Beginning 0.02 radStep Size 0.004 rad

JLP (Cost) Threshold 0.01

α Minimum 0.1 rad/sβ Minimum 0.1 rad/s

‖ωr‖2 Minimum 0.01

Initial Axes (Spherical) φ0

[45◦ 45◦ 90◦ 45◦

]

0 5 10 15 20 25 300

0.05

0.1

0.15

JLP ≤ Threshold

t[s]

Cos

tV

alue

Fig. 6: The values of the cost function J ( ), the lowpass-filtered costfunction JLP ( ) and the threshold ( ) are shown for the first trial.

imitating to grab a door knob, turning the knob and openingthe door. The motion was selected to provide a realistic day-to-day movement which excites FE as well as PS.

For the first three trials, the sensors were placed as depictedin Fig. 1. The sensors were approximately aligned with therotation axes, meaning that the z-axis of IMU1 was alignedwith the FE axis and the x-axis of IMU2 with the PS axis.Subsequently, a more arbitrary sensor position and orientationplacement was used for trials four to six (see Fig. 4).

For all experiments, the exact same set of parameterspresented in Tab. I was used.

III. RESULTS

Six trials were carried out, trials 1-3 with the first IMUplacement (Fig. 1) and trials 4-6 with the second (Fig. 4).

The estimated elbow joint axes for all trials are presentedin Fig. 5. For all trials, the algorithm was started at t = 0.

The joint angles obtained by the optical reference aredisplayed with the angles computed by the proposed methodfor all trials in Fig. 7. Because the IMU-based angles are basedon the “zero pose” and the optical angles are absolute, theoptical angles were shifted to match the IMU-based angles3.

The first three trials were conducted with assumed IMU-to-segment alignment. As mentioned in the introduction, oneof the most common methods to obtain joint angles is byaligning the IMUs and using predefined fixed rotation axes.

3By choosing a pose identical to the absolute zero, this can be avoided.However, slight offsets from reference system are to be expected.



5

−1

0

1E

st. A

xis

aTrial 1

0 5 10 15 20 25 30−1

0

1

t[s]

Est

. Axi

sb

Trial 2

0 5 10 15 20 25 30

Trial 3

0 5 10 15 20 25 30

−1

0

1

Est

.Axi

sa

Trial 4

0 5 10 15 20 25 30−1

0

1

t[s]

Est

. Axi

sb

Trial 5

0 5 10 15 20 25 30

Trial 6

0 5 10 15 20 25 30

Fig. 5: The estimated rotation axes for flexion/extension aA in the frame of IMU1 and pronation/supination bB in the frame of IMU2 are shown for all sixtrials x( ), y( ) and z ( ). For the first three trials, the assumed axes from the IMU alignment are indicated by horizontal lines ( ). The times whenJLP crosses the threshold are indicated by vertical lines ( ). The motion was started at 0 s.

020406080

100

Flex

ion/

Ext

ensi

onA

ngle

[◦]

Trial 1

0 5 10 15 20 25 30

−200

20

40

60

t[s]Pron

atio

n/Su

pina

tion

Ang

le[◦]

Trial 2

0 5 10 15 20 25 30

Trial 3

0 5 10 15 20 25 30

020406080

100

Fle x

ion/

Ext

ensi

onA

ngle

[◦]

Trial 4

0 5 10 15 20 25 30

−200

20

40

60

t[s]Pron

atio

n/Su

pina

tion

Ang

le[◦]

Trial 5

0 5 10 15 20 25 30

Trial 6

0 5 10 15 20 25 30

Fig. 7: The flexion/extension and pronation/supination angles acquired by the optical reference measurement ( ) and by using the IMU measurements basedon the estimated axes ( ) are shown for all six trials. The times when JLP crosses the threshold are indicated by vertical lines ( ).



6

020406080

Fle x

ion/

Ext

ensi

onA

ngle

[◦]

0 5 10 15 20 25 30

0

20

40

60

t[s]Pron

atio

n/Su

pina

tion

Ang

le[◦]

Fig. 8: The flexion/extension and pronation/supination angles acquired by theoptical reference measurement ( ) and by using the IMU measurementswith manually chosen fixed axes known through the careful alignment of theIMU sensors ( ).

TABLE II: RMS errors of the joint angles, comparing the joint angles acquiredby the IMU measurements and the angles provided by the optical referencefor the six trials. The average RMS values are presented in the last column.The RMS values were calculated from 10 s after the start of the movementup to the end of the measurement.

Sensor Position 1 Sensor Position 2Trial 1 2 3 4 5 6 x

Using the Proposed AlgorithmFlexion/extensionRMS Error

1.7◦ 1.3◦ 1.7◦ 5.4◦ 2.5◦ 3.7◦ 2.7◦

Pronation/supinationRMS Error

3.9◦ 6.6◦ 3.5◦ 2.8◦ 2.5◦ 3.6◦ 3.8◦

Using Manual Alignment of the SensorsFlexion/extensionRMS Error

3.9◦ 3.1◦ 4.4◦ 3.8◦

Pronation/supinationRMS Error

8.2◦ 8.8◦ 9.2◦ 8.7◦

For the first IMU placement (Fig. 1) the axes are assumed tobe[0 0 1

]Tfor IMU1 and

[1 0 0

]Tfor IMU2. Fig. 8

shows the joint angles of the first trial using the assumed fixedaxes.

The cost function derived in (7) and its lowpass filteredcounterpart can be seen for the first trial in Fig. 6.

For a quantitative measure of the error, the root mean square(RMS) values of the difference between the reference and themeasurement were determined for both joint angles and alltrials, see Tab. II. The values represent the time from 15 safter start to the end of the trial, giving the algorithm time toconverge but at the same time penalizing convergence timeslonger than 15 s. Additionally, the RMS errors for the assumedfixed axes of position 1 are presented.

IV. DISCUSSION

In all experiments, the estimated axes converged to a steadyvalue. By looking at Fig. 5 it becomes apparent that the timeof convergence is consistent with the time of the stepsizeswitchover indicated by the vertical lines for all trials, apartfrom the second trial. When the second trial is assumed to havea convergence time of 20 s, the overall average convergencetime results in 9.5 s. After the convergence a slight fluctuationof the estimated axes can be observed. This fluctuation isassumed to be caused by the skin motion artifacts mentionedearlier. A smaller step size could decrease the fluctuations,

the disadvantage, however, would be that potential estimationerrors are corrected at a slower rate.

For each estimated axis, there are two correct solutionswhich are the positive and the negative real rotation axis.For both sensor positions, all estimated axes converged tothe identical solution and not the negated counterpart. Sincethe same initial axes were used for all trials and due tothe relatively moderate stepsize, it is not surprising that thesolution closer to the initial axes was found. A remainingproblem is still, that a change in the sign of the estimatedaxis leads to a change in the sign of the correspondingestimated angle. This means that with the presented version itis not guaranteed that the estimated angles have the expecteddirection. A simple solution for this problem is to define the“zero pose” on an extreme joint position and to define the jointangles to be positive. This would mean that if we measure jointangles below a negative threshold value we can simply switchthe sign of either the estimated angle or axis.

The switching of the step size and the convergence of thegradient descent can be observed in Fig. 6 for the first trial. Inthe first 5 s it can be clearly seen how the algorithm descendsto the valley of the cost function. As soon as the lowpass-filtered curve of the cost function arrives at the threshold, thestepsize is permanently switched from the high initial valueto a low one. After the switch both of the estimated axesbecome much smoother (see Fig. 5). However, the error doesnot stay below the threshold but fluctuates within relativelyhigh bounds. By looking at the axes plot for trial one, we cansee that the axes already converged after approximately 5 s.This means that the high cost function values do not originatefrom badly estimated axes. A possible explanation is a slightmovement of the first IMU when the elbow is close to fullflexion. These results underline how big the influence of skinmotion artifacts are. The two-stage switching implementationof the gradient descent algorithm is capable of steadying theaxes estimation without sacrificing the speed of convergence.

The ultimate goal is not to estimate axes but to obtainprecise joint angles in real-time. Hence, the most significantfigure is the comparison of the obtained angles to the opticalreference angles (Fig. 7). The proposed method was capableof tracking the elbow angles for all six trials. After theconvergence of the estimated axes, the angles followed thereference almost perfectly. In the second trial, the switchoverto the smaller stepsize happened too soon before the actualconvergence. This resulted in a very slow convergence. Asa result, the PS angle is off until approximately 20 s intothe measurement. Note that a single bad axis estimationcan influence both joint angles due to the nature of theEuler decomposition. The presented RMS values for eachangle and trial (Tab. II) provide a meaningful performanceindex for the proposed method. Note that not only errors inthe estimated axes can lead to a deviation from the opticalreference angles. Other sources like the measurement errorsof the optical tracking system, the inaccuracies of derivingcoordinate systems from bony landmarks, the inaccuracies ofthe estimation of the glenohumeral rotation center, the errorsof the IMU measurements, the errors of IMU sensor fusionalgorithm and most importantly the skin motion artifacts of



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both, IMUs and optical markers have to be considered as well.As stated in the introduction, a common way of measuring

joint angles using IMUs is to use careful alignment of theIMUs. In the first three trials, we used an IMU placementwhere the IMUs were approximately aligned with the jointaxes (see Fig. 1). Hence, the results of the common methodare a good performance index for the presented method. Both,evaluating Fig. 8 as well as the RMS errors (Tab. II) showsagain how hard it is to properly align the sensors and thatsmall mistakes in the alignment have a strong impact on themeasurement quality. In Fig. 5 a significant difference betweenthe estimated PS axes and the assumed PS axis from the sensorplacement is evident. The reason for this is that the orientationof the PS axis is hard to spot with the naked eye. Thus, manualsensor alignment along the PS axis easily compromises themeasurement accuracy.

Other studies have evaluated their IMU-based 2DOF anglemeasuring algorithms on mechanical devices and on the humanbody. For tests on mechanical devices using rotational en-coders, typical RMS errors varied between 1 and 4◦ dependingon the method used and on the speed of motion [19], [20]. Fortests on actual human elbows, RMS errors of 2.4◦ (FE) and4.8◦ (PS) were measured with respect to an optical marker-based system in [2]. This is very comparable to the RMS errorsof the presented method of 2.7◦ (FE) and 3.8◦ (PS). Thus,the presented method does not compromise accuracy for thegained benefit of an automatic sensor-to-segment calibration.

V. CONCLUSION

A method that allows the determination of joint angleswithout the need of specific IMU-to-segment alignment wasdeveloped. The user is allowed to attach both sensors withan arbitrary orientation. After the user assumes a zero pose,the algorithm automatically calibrates online using the naturalmovements of the user. Accurate joint angles are obtainedwithin 9.5 s on average.

An optical marker-based tracking system was used to eval-uate the performance of the method on a human subject.The joint angles could be accurately tracked with the sameaccuracy range as the state-of-the-art methods (which needeither careful sensor alignment or predefined calibration move-ments). The method proved to be robust enough to deal withthe disturbances of skin motion and quickly converged to thecorrect joint axes.

Further studies should include the evaluation in a clinicalsetting and also a bigger quantity of subjects. Here, two ques-tions will be answered: is the motion of the patient group (e.g.stroke patients) rich enough to guarantee convergence and isthere an increase in the skin motion artifacts in the patientgroup? Another important aspect is an analytical investigationof the local and global minima of the nonlinear cost function.For the implementation we used a first-order gradient descentmethod, using a second-order approach such as the Gauss-Newton method could lead to faster convergence.

The proposed method is designed to work with any two-dimensional joint. Therefore, it should be investigated whetherthe method can be applied to other joints of the human bodythat can be simplified as two-dimensional (e.g. the ankle joint).

ACKNOWLEDGMENT

We thank our colleagues of the Fraunhofer IKP (Pascal-straße 8-9, 10587 Berlin, Germany) for providing the opticalmeasurement system and helping us with the measurements.

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alignment-free, self-calibrating elbow angles measurement ......kinematics from imu data, is that...

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